We only need to prove the Triebel-Lizorkin case, i.e. \({A}={F}\) . Indeed suppose we get the case \({A}={F}\) , then by real interpolation (), for each \(0<p,q\le \infty \)  and \(s\in \mathbb {R}\)  there is a \(C_{p,q,s}>0\)  such that (see [1]}, in fact \(C_{p,q,s}=C_q\) )
\(\Vert T\Vert _{{B}_{pq}^s(\mathbb {R}^n)\rightarrow {B}_{pq}^s(\mathbb {R}^n)}\le C_{p,q,s}(\Vert T\Vert _{{F}_{pp}^{s-1}(\mathbb {R}^n)\rightarrow {F}_{pp}^{s-1}(\mathbb {R}^n)}+\Vert T\Vert _{{F}_{pp}^{s+1}(\mathbb {R}^n)\rightarrow {F}_{pp}^{s+1}(\mathbb {R}^n)}),\) 
whenever \(T:{F}_{pp}^{s-1}(\mathbb {R}^n)\rightarrow {F}_{pp}^{s-1}(\mathbb {R}^n)\)  is a bounded linear map such that \(T:{F}_{pp}^{s+1}(\mathbb {R}^n)\rightarrow {F}_{pp}^{s+1}(\mathbb {R}^n)\)  is also bounded. Taking \(T=\vartheta ^r\)  and replacing \(M_{p,q}^{n,s}\)  (obtained from the \({F}\) -cases) with \(\max (M_{p,q}^{n,s},M_{p,p}^{n,s-1},M_{p,p}^{n,s+1})\)  we complete the proof.
Step 1: The case \(n=1\) , \(0<p<\infty \) , \(0<q\le \infty \)  and \(s>\frac{1}{p}-1\) . We use [2]}.
Recall \(\lambda =(\lambda _j)_{j=0}^\infty \)  from Definition REF . When \(s>\max (0,1-\frac{1}{p})\) , by [3]}, we have \(\Vert f\Vert _{{F}_{pq}^s(\mathbb {R})}\approx _{p,q,s}\Vert f\Vert _{L^p(\mathbb {R}^n)}+\big \Vert x\mapsto \big (\int _0^\infty t|(\lambda _1(\frac{\cdot }{t})\ast f)(x)|\frac{dt}{t^{sq+1}}\big )^{1/q}\big \Vert _{L^p(\mathbb {R})}\) . The same dilation argument from [2]} shows that
\(\Vert \vartheta ^rf\Vert _{{F}_{pq}^s(\mathbb {R})}\approx |r|^{-\frac{1}{p}}\Vert f\Vert _{L^p}+|r|^{s-\frac{1}{p}}\Big \Vert x\mapsto \Big (\int _0^\infty |(\lambda _1(\tfrac{\cdot }{t})\ast f)(x)|\frac{dt}{t^{sq}}\Big )^\frac{1}{q}\Big \Vert _{L^p}\lesssim _{p,q,s}(|r|^{-\frac{1}{p}}+|r|^{s-\frac{1}{p}})\Vert f\Vert _{{F}_{pq}^s(\mathbb {R})}.\) 
In particular we can take \(M_{p,q}^{1,s}:=\max (\frac{1}{p},s-\frac{1}{p},\frac{1}{p}-s)\)  in this case.
Step 2: The case \(n\ge 2\) , \(0<p<\infty \) , \(0<q\le \infty \)  and \(s>n\max (0,\frac{1}{p}-1,\frac{1}{q}-1)\) . We use Proposition REF  REF :
\(\Vert \vartheta ^rf\Vert _{{F}_{pq}^s(\mathbb {R}^n)}\approx &\Big (\int _{\mathbb {R}^{n-1}}\Vert \vartheta ^r(f(x^{\prime },\cdot ))\Vert _{{F}_{pq}^s(\mathbb {R})}^pdx^{\prime }\Big )^{1/p}+\Big (\int _{\mathbb {R}}\Vert f(\cdot ,rx_n)\Vert _{{F}_{pq}^s(\mathbb {R}^{n-1})}^pdx_n\Big )^{1/p}\\\lesssim &(|r|^{-\frac{1}{p}}+|r|^{s-\frac{1}{p}})\Big (\int _{\mathbb {R}^{n-1}}\Vert f(x^{\prime },\cdot )\Vert _{{F}_{pq}^s(\mathbb {R})}^pdx^{\prime }\Big )^{\frac{1}{p}}+|r|^{-\frac{1}{p}}\Big (\int _{\mathbb {R}}\Vert f(\cdot ,y_n)\Vert _{{F}_{pq}^s(\mathbb {R}^{n-1})}^pdy_n\Big )^{\frac{1}{p}}\\\lesssim &(|r|^{-M_{p,q}^{1,s}}+|r|^{M_{p,q}^{1,s}})\Vert f\Vert _{{F}_{pq}^s(\mathbb {R}^n)}.\) 

In particular we can take \(M_{p,q}^{n,s}:=M_{p,q}^{1,s}\)  in this case.
Step 3: The case \(0<p<\infty \) , \(0<q\le \infty \)  and \(s\in \mathbb {R}\) . We use Proposition REF  REF .
Let \(m\ge 0\)  be such that \(s+m>n\max (0,\frac{1}{p}-1,\frac{1}{q}-1)\) . Note that \(\vartheta ^\frac{1}{r}\circ \partial ^\alpha =r^{\alpha _n}\partial ^\alpha \circ \vartheta ^\frac{1}{r}\) , thus
\(&\Vert \vartheta ^rf\Vert _{{F}_{pq}^s(\mathbb {R}^n)}\approx \textstyle \inf \Big \lbrace \sum _{|\alpha |\le m}\Vert g_\alpha \Vert _{{F}_{pq}^{s+m}(\mathbb {R}^n)}:\vartheta ^rf=\sum _{|\alpha |\le m}\partial ^\alpha g_\alpha \Big \rbrace \\=&\inf \Big \lbrace \sum _{|\alpha |\le m}\Vert g_\alpha \Vert _{{F}_{pq}^{s+m}(\mathbb {R}^n)}:f=\sum _{|\alpha |\le m}r^{\alpha _n}\partial ^\alpha (\vartheta ^\frac{1}{r}g_\alpha )\Big \rbrace =\inf \Big \lbrace \sum _{|\alpha |\le m}\frac{\Vert \vartheta ^rh_\alpha \Vert _{{F}_{pq}^{s+m}(\mathbb {R}^n)}}{|r|^{\alpha _n}}:f=\sum _{|\alpha |\le m}\partial ^\alpha h_\alpha \Big \rbrace \\\lesssim &(|r|^{-m}+1)(|r|^{-M_{p,q}^{n,s+m}}+|r|^{M_{p,q}^{n,s+m}})\Vert f\Vert _{{F}_{pq}^s(\mathbb {R}^n)}.\) 

In particular we can take \(M_{p,q}^{n,s}:=M_{p,q}^{n,s+m}+m\)  in this case.
Step 4: The case \(p=\infty \) , \(0<q\le \infty \)  and \(s\in \mathbb {R}\) . We use Proposition REF  REF .
For \(x\in \mathbb {R}^n\)  and \(R>0\) , we see that \(f\in {F}_{pq}^s\big (B\big ((x^{\prime },\frac{x_n}{r}),\max (R,\frac{R}{|r|})\big )\big )\)  implies \(\vartheta ^r f\in {F}_{pq}^s(B(x,R))\) . Therefore
\(&\Vert \vartheta ^rf\Vert _{{F}_{\infty q}^s(\mathbb {R}^n)}\approx \sup _{x\in \mathbb {R}^n,R>0}R^{-\frac{n}{q}}\inf \big \lbrace \Vert \tilde{g}\Vert _{{F}_{qq}^s(\mathbb {R}^n)}:\tilde{g}|_{B(x,R)}=(\vartheta ^rf)|_{B(x,R)}\big \rbrace \\\le &\sup _{x\in \mathbb {R}^n,R>0}R^{-\frac{n}{q}}\inf \big \lbrace \Vert \vartheta ^\frac{1}{r}\tilde{f}\Vert _{{F}_{qq}^s(\mathbb {R}^n)}:\tilde{f}\big |_{B\big ((x^{\prime },\frac{x_n}{r}),\max (R,\frac{R}{|r|})\big )}=f\big |_{B\big ((x^{\prime },\frac{x_n}{r}),\max (R,\frac{R}{|r|})\big )}\big \rbrace \\\lesssim &(|r|^{-M_{q,q}^{n,s}}+|r|^{M_{q,q}^{n,s}})\sup _{x\in \mathbb {R}^n,R>0}R^{-\frac{n}{q}}\Vert  f\Vert _{{F}_{qq}^s\left(B\left((x^{\prime },\frac{x_n}{r}),\max (R,\frac{R}{|r|})\right)\right)}\\\lesssim &(|r|^{-M_{q,q}^{n,s}}+|r|^{M_{q,q}^{n,s}})(1+|r|^{-\frac{n}{q}})\sup _{x\in \mathbb {R}^n,R>0}R^{-\frac{n}{q}}\Vert  f\Vert _{{F}_{qq}^s(B(x,R))}\lesssim (|r|^{-M_{q,q}^{n,s}-\frac{n}{q}}+|r|^{M_{q,q}^{n,s}})\Vert  f\Vert _{{F}_{\infty q}^s(\mathbb {R}^n)}.\) 

Taking \(M_{\infty ,q}^{n,s}:=M_{q,q}^{n,s}+n/q\) , we complete the proof.
